Ending free movement after Brexit may not result in lower migration to the UK, according to a Lords report.
Ministers are yet to outline the details of the proposed immigration system once the country has departed the European Union.
But the scheme is expected to impose restrictions on free movement rules as the Government attempts to reduce overall net migration to the tens of thousands.
However, a report from the House of Lords EU Home Affairs Sub-Committee raises doubts about the impact of any restoration of national control over EU migration.
It points out that, until the referendum, net migration - the difference between the numbers arriving to live in the country and those leaving - was consistently higher from outside the EU.
This is despite routes from outside Europe being covered by a number of restrictions.
The wide-ranging assessment says: "Restoration of national control over EU migration may or may not, therefore, deliver a reduction in overall net migration."
In the most recent official figures, covering the year to the end of September, it was revealed that both immigration and net migration from the EU were higher than that for the rest of the world for the first time. Overall net long-term international migration was estimated to be 273,000.
The committee also concluded that cutting EU immigration is unlikely to provide a "quick fix" for low wages.
Factors such as the National Minimum Wage, National Living Wage and inflation are more significant in driving or impeding real wage growth for low earners, the report argues.
It also warns that extending the work permit system currently used for non-EU nationals to those from the union would disproportionately affect some employers' ability to sponsor EU workers, and could result in labour shortages in areas including the NHS and horticulture.
It says any system that is "hedged" with exemptions for particular sectors and schemes could produce the "worst of all worlds, failing to deliver a meaningful reduction in immigration while also proving more onerous and costly for employers, prospective applicants, and those charged with enforcement".
The report endorses the pursuit of a "two-way agreement" with the EU on future migration flows, and suggests that offering preferential treatment to EU nationals in the UK's future immigration regime could increase the likelihood of securing a reciprocal approach to UK nationals in the EU.
Baroness Prashar, chair of the committee, said: "The precise manner in which the Government proposes to 'end' free movement is a pivotal aspect of the United Kingdom's approach to negotiations with the European Union and could have far-reaching consequences for the UK's future trading relationship with the EU.
"Crucial sectors of the economy depend on EU migrant labour, so it is essential that any changes don't endanger the vibrancy of the UK economy.
"We therefore recommend a phased transition to avoid short-term shocks to particular sectors."
She added that the committee was "struck by the weaknesses and gaps in the UK's migration statistics".
A Home Office spokesman said: "Once we have left the European Union it will be the Government that sets our immigration rules.
"We are currently considering the various options as to how EU migration might work once we have left and it would be wrong to set out further positions at this stage."